Solve the inequality
$$\dfrac{2x^4+2x^2}{\sqrt{x+1}}+(x+2)\sqrt{x+1}>x ^3 + 2x^2 + 5x.$$
I tried. By putting $t = \sqrt{x+1}$, we have
$$2t^8-t^7-8t^6+t^5+15t^4-4t^3-11t^2+4t+4>0.$$
Using Maple, I got
$$(2t^2-t-2)(t^2-t-1)(t^4+t^3-t^2-t+2)>0.$$
How to solve the given inequality with another way?